# Rational maps are $d$-adic Bernoulli

### Abstract

Freire, Lopes and Mañé proved that for any rational map $f$ there exists a natural invariant measure $\mu_f$ [5]. Mañé showed there exists an $n>0$ such that $(f^n,\mu_f)$ is measurably conjugate to the one-sided $d^n$-shift , with Bernoulli measure $(\frac{1}{d^n},\ldots,\frac{1}{d^n})$ [15]. In this paper we show that $f,\mu+f)$ is conjugate to the one-sided Bernoulli $d$-shift. This verifies a conjecture of Freire, Lopes and
Mañé [5] and Lyubich [11].

## Authors

Deborah Heicklen

Christopher Hoffman